self studying advice on analysis

Here is some advice on self-studying* analysis. Much of it is applicable to independent study of pure mathematics in general.

1. No matter what part of mathematics you are trying to learn, a lot of what you will learn -- especially at first -- is how to read a mathematics text. This is really a nontrivial skill. You can't read a math text like a novel, i.e., flipping the page every minute or two and concentrating more on the thrill of the ride and the drama of the final destination than the page-by-page details. (Admittedly there are a few novels that one cannot read like a novel, but never mind that...) Most independent students realize this very quickly and are then tempted to try the other extreme: i.e., insisting on not moving on to the next page until the current page is fully understood. This is a bad strategy as well: it is extremely inefficient and will cause unnecessary frustration, as it is in the nature of reading anything that often reading a little bit farther can have a clarifying effect. (As a professional research mathematician I (independently!) read lots of mathematics, and sometimes it still happens to me that I get stuck on a particular sentence for a while, not understanding how to verify the claim that it makes. Sometimes this explanation comes in the next sentence, the next paragraph or the next page. That's bad writing, but almost every writer does it once in a while, I think.)

Instead you need to find the interior maximum: i.e., the right balance between reading ahead and then dipping back behind. I think this must be different for everyone, but I'll just mention one good strategy: if you're skipping something, it's good to fix in your own mind -- and even writing it down couldn't hurt -- what it is that you're skipping. For instance maybe you don't want to read the two page proof of Theorem X.Y at the moment. So you read on to see whether it's reasonable to take that proof as a "black box". Sometimes it is and sometimes it isn't.

By the way, I am coming to believe that "interior maximum" is a good maxim to repeat to students of mathematics. We mathematically minded people tend to be very idealistic and want to go to extremes, and just as in calculus this is usually not optimal. In your question for instance you say that you "try to solve each and every problem". That may well be too much: it depends on how much time you spend. When you're reading and learning independently you need to strike a balance between actually being independent and working things out for yourself and not spending an inordinate amount of time on some point that you know you could look up the answer to if you so chose.

2. If you find that you can solve most of the problems in a reasonable amount of time, that sounds great to me and maybe you just need to be happier with yourself and your progress. It is also a very positive feature that you are solving problems for which you can look up the solutions when you need to (and also that you seem to have enough discipline not to look at the solution after 5 minutes; that's a huge problem with independent study).

Concerning written solutions to problems, here's a dirty secret: when they are included at all, they are rarely written up with anything like the care with which the text itself is written. (For mainstream textbooks, often there is a separate "student solution guide" and often this was not primarily prepared by the author of the text.) For Spivak's Calculus for instance, I am very familiar with the answer book from time grading and then teaching courses out of this text. Whenever more than one student turned in an essentially correct but unnecessarily eccentric solution, I would look in the answer book and often find the solution there. It is also the case that most problems have more than one solution, and if you have the task of writing up solutions to hundreds of problems, then as you might imagine you might optimize for the shortest solution or simply the first one that pops into your head rather than the one(s) which you feel are most instructive.

As an example, it seems to me that the problem you mention about $x_n \rightarrow L \neq 0$, $x_n y_n \rightarrow M \implies y_n \rightarrow \frac{M}{L}$ has a pretty straightforward solution: we establish, if we haven't done so earlier, that $\frac{1}{x_n} \rightarrow \frac{1}{L}$, and then use the fact that the limit of the product is the product of limits.

3. Here's another piece of advice for the independent student: try to have at least one backup text to consult if you don't like the exposition in your primary text. Some texts leave very basic things to the reader as exercises. There is a philosophy behind this, but I don't really agree with it: I think that the student of a subject deserves to see complete proofs of the basic facts. Filling in some of these independently could be instructive if the student succeeds, but what if they fail or simply get discouraged? Having exercises for the student to solve is very good, but in most subjects in mathematics there are essentially infinitely many exercises to be assigned at any level. It feels a bit lazy to me to assign proofs of the stated theorems as exercises. This is where it's advantageous to have multiple texts: what is stated as an exercise in one may well be proved in another.

Maybe for this it helps to have one source which you can be fairly sure will include proofs of all the basic things, whatever its other merits. For elementary real analysis, I might suggest these lecture notes of mine, in which a "more is more" approach to proof is generally taken.

*: Though it is a very standard term on this site and I understand what it means -- one is not taking a course or being expressly guided by an experienced practitioner -- the term "self-studying" has never sat particularly well with me. I tolerate it, but the related term "self-learning" makes me want to retch: what other kind of learning is there? Anyway, maybe unassisted study or independent study are better terms here.

Don't sweat it if you get a few problems wrong, especially if you worked on them for a while before looking up the solution. There are plenty of good problems in analysis to try, just keep trying new ones, seeing what you can do, then seeing how an expert solves it (i.e. looking at a solution). Eventually you will pick up the tricks and you'll be able to solve the problems on your own!

You should move on as soon as you mostly understand the solutions to all the problems in a section, even if you didn't come up with those solutions on your own. Then, continually come back to sections you didn't understand before - you'll notice that old material will seem easier the second or third time around.

I also highly recommend getting a second (or third) book to harvest problems and ideas from. Also, keep hanging around math.stackexchange and you'll probably pick up a few tricks here and there =)